Capacitors

1. Core Relations

• Charge: Q = C · V
• Current–voltage relationship: i(t) = C · (dv/dt)
• Energy stored: E = ½ · C · V²

The Intuition

A capacitor opposes changes in Voltage (just like an Inductor opposes changes in Current). Ideally, it acts like a frequency-dependent resistor.

2. Physics & Operation

Charge builds up on the plates, creating an Electric Field.

1. The capacitor charges until the voltage across its plates equals the source voltage.
2. Once fully charged (no difference in potential), electrons stop moving. Current = 0.

Energy is stored in its electric field, so ideally no power loss. In a real world application there is some power from the ESR and dielectric loss.

Interactive: DC Block vs AC Pass

See how the potential difference drives current flow.

DC: Current flows slowly to charge plates. Once charged, current stops.

3. Frequency Behavior

In the frequency domain, a capacitor is an Impedance.

• Impedance Formula: Z = 1 / (jωC)
• Magnitude: |Z| = 1 / (2π · f · C)

The Rules of Thumb:

• DC (f=0): Z is Infinite. Acts like an Open Circuit.
• High Freq: Z approaches 0. Acts like a Short Circuit (Wire).

Parasitics (ESL)

Real capacitors have "Parasitic Inductance" (ESL) from their leads and internal construction. At very high frequencies, the ESL takes over and the capacitor starts looking like an Inductor, effectively stopping it from acting as a short circuit.

Phase Relationship ("ICE")

Unlike a resistor (where Voltage and Current happen instantly together), a capacitor takes time to charge.

• Current flows First: You have to push current into the capacitor before the voltage can rise.
• Voltage Lags: The voltage reaction is delayed by 90° (¼ of a cycle).

Mnemonic: "ELI the ICE man"

• E = Voltage (EMF)
• I = Current
• L (Inductor): E leads I (ELI)
• C (Capacitor): I leads E (ICE) -> Current comes before Voltage

Phase: I leads V by 90°

Current (Red) is already at its peak when Voltage (Yellow) is just starting at zero.

Why this matters: This 90° phase shift is why capacitors are used to stabilize feedback loops (compensation) and why they determine the "Phase Margin" in power supplies.

4. Common Uses

AC Coupling (Blocking)

A series capacitor placed between two stages.

• Goal: Pass the AC signal (Audio/RF) but block the DC bias voltage.
• Allows a 3.3V microcontroller to talk to a 5V amplifier without blowing up the DC bias points.

Bypass / Decoupling

A capacitor placed from a Supply Pin to Ground.

• Filters out any unwanted AC signal from the power supply to ground.
• Goal: Act as a bridge for AC only.
• Layout: Place small ceramics as close to the pin as possible. Minimize loop area.

Bulk Capacitance

Large capacitors (usually Electrolytic) on the main power rail.

• When the IC switches and demands a sudden gulp of current, the capacitor provides it immediately (since the main power supply is too far away and has trace inductance).
• Goal: Handle large, slow power surges and smooth out low-frequency ripple.

5. Dielectric Types

Type	Dielectric	Best For	Notes
Ceramic	C0G / NP0	Filters, Timing, RF	The Gold Standard. Temp stable, no piezo noise, but low capacitance values.
Ceramic	X7R / X5R	Decoupling, Bulk	High capacitance density. Warning: Capacitance drops with DC bias.
Film	PP / PET	Audio, High Voltage	Very low ESR, linear, handles high voltage spikes well. Large physical size.
Electrolytic	Aluminum	Bulk Energy Storage	High Capacitance per dollar. High ESR, dries out over time.

6. RC Time Constants

When you charge a capacitor through a resistor, it doesn't happen instantly. It follows a curve.

• Time Constant: τ (tau) = R · C
• Charge Equation: v_C(t) = V_S · (1 - e^-t/τ)

Key Milestones:

• 1τ: 63% Charged
• 3τ: 95% Charged
• 5τ: ~99% Charged (Considered "Full" in engineering)

7. Differentiators & Integrators

By arranging the C and R, we can perform calculus on signals.

How it works

• At high frequencies, Reactance is low, Capacitor acts more like a wire, and at low frequencies, the capacitor is acting more like an open, blocking DC.
• As the frequency of the input increases, the output looks more and more like the input. Basically like a high pass filter.
• The voltage of the resistor is dependent on the current released by the capacitor. The current released by the capacitor is determined by I_C = C · (dV / dt), so V_out = R · C · (dV_in / dt), which means the output voltage is the derivative of the input.

Differentiator (High-Pass Filter)

Output is proportional to the rate of change of the input.

RC Differentiator (High-Pass)

• Function: Blocks DC, passes fast edges.
• Use Case: Rising Edge Detector. A square wave input becomes sharp spikes at the output.
• Math: V_out ≈ RC · (dV_in/dt)

Integrator (Low-Pass Filter)

Output is proportional to the accumulation of the input.

Similarly

• Integrator acts similarly. Acts as a low pass filter for a sinewave.
• Acts as an integrator for other waves.
• I_C = C · (dV / dt) → V_OUT = ∫ I_C dt × (1/C).
• I_C = I_R = V_IN / R → (1 / RC) times the integral of V_IN is your output.

RC Integrator (Low-Pass)

• Function: Smooths out fast signals, passes DC.
• Use Case: Averaging a PWM signal to create a DC voltage (DAC).
• Math: V_out ∝ ∫ V_in dt.

8. Real-World Parasitics (ESR & ESL)

An ideal capacitor is purely capacitive. A real capacitor is actually a resistor, inductor, and capacitor in series. This changes how we design power supplies and high-speed circuits.

ESR (Equivalent Series Resistance)

The internal resistance of the plates, leads, and dielectric losses.

• Heat Generation: Ripple current flowing through the ESR creates heat (P = I² · ESR). High ESR caps (like Electrolytics) can overheat and pop if the ripple current is too high.
• Voltage Ripple: In power supplies, output ripple is often determined by V_ripple = I_load × ESR. Lower ESR = Cleaner Power.
• Stability: Some older LDO regulators rely on a tiny bit of ESR to stay stable. If you replace an old Tantalum cap with a modern Ceramic (near-zero ESR), the LDO might oscillate.

ESL (Equivalent Series Inductance)

The inductance of the leads and internal structure.

• Frequency Limit: Inductance opposes high-frequency changes.
• Package Size Matters: A large 1206 capacitor has higher ESL than a tiny 0402 capacitor. This is why we place small 0402/0201 caps closest to the IC pins—they respond faster.

The Impedance "V" Curve (Self-Resonance)

Because of ESL, a capacitor only acts like a capacitor up to a certain frequency called the SRF (Self Resonant Frequency).

1. Down Slope (Capacitive): Impedance drops as frequency rises. This is normal behavior.
2. The Bottom (Resistive): The impedance hits its lowest point. This value is the ESR.
3. Up Slope (Inductive): Above the SRF, the ESL takes over. The impedance starts rising again. The capacitor is now effectively an Inductor and blocks high-frequency noise less effectively.

Design Tip: Parallel Caps

This is why we place a 10µF (Bulk) and 0.1µF (Ceramic) in parallel.

• The large cap handles low frequencies.
• The small cap handles high frequencies (because it has lower ESL/higher SRF). Together, they cover a wider frequency range than either could alone.

10. Non-Ideal Properties

Real capacitors are not just "C". They are complex devices with leakage, memory, and physical sensitivity. Ignoring these will break precision circuits.

A. Dielectric Absorption ("Soakage")

If you fully charge a capacitor, then short it out to 0V for a second, then open the circuit... the voltage will creep back up!

• The Physics: Charge gets "trapped" deep inside the dielectric material and takes time to release.
• The Consequence: This ruins precision Sample-and-Hold circuits and long-period Integrators. The capacitor "remembers" its previous voltage history.
• The Fix: Never use Electrolytic or High-K Ceramic (X7R) for precision timing. Use Polypropylene (PP) or Polystyrene film capacitors.

B. Piezoelectric Effects ("Microphonics")

Multi-layer Ceramic Capacitors (MLCCs) using High-K dielectrics (X7R, Z5U, Y5V) are piezoelectric.

1. Microphone Effect: If you tap the PCB, the capacitor generates a voltage spike. (Bad for high-gain audio preamps or vibration-sensitive sensor circuits).
2. Speaker Effect: If you apply an AC signal (audio frequency), the capacitor physically vibrates. This causes "Singing Capacitors" (audible whining) in power supplies.

• The Fix: Use C0G / NP0 (Class 1) ceramics or Film capacitors in sensitive signal paths. They are not piezoelectric.

C. Leakage (Insulation Resistance)

Every capacitor has a parallel resistor (R_leak) that slowly discharges it.

• Electrolytics: Very leaky. You cannot use them for timers longer than a few seconds.
• Ceramics/Film: Extremely low leakage (pico-Amps). Good for minutes or hours of hold time.

D. Detailed Film Capacitor Selection

"Film" is too broad. The Art of Electronics distinguishes them carefully:

Dielectric	Symbol	Properties	Best Application
Polyester	PET / Mylar	Cheap, generic. High soakage.	General coupling/decoupling where precision doesn't matter.
Polypropylene	PP	Low loss, Low soakage.	Precision timing, high-power pulses, high-end audio.
Polystyrene	PS	The old "King of Precision."	Extremely stable. Hard to find now (replaced by C0G/PP). Melts easily when soldering!
Polycarbonate	PC	Good temp stability.	Mostly obsolete/rare now.

E. The "Parallel Caps" Logic

Why do engineers place a 10µF Tantalum and a 0.1µF Ceramic in parallel? Why not just use one 10.1µF?

Reason: ESL (Equivalent Series Inductance).

• The 10µF cap has high ESL. Above ~1MHz, it stops being a capacitor and becomes an inductor (Open circuit to high freq noise).
• The 0.1µF cap has low ESL. It stays capacitive up to ~100MHz, filtering the noise the big cap missed.

Design Rule

In high-speed digital circuits, Physical Size matters more than value. A small 0402 0.1µF capacitor has lower inductance (and filters better at high GHz) than a large 1206 0.1µF capacitor.